Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website. See our User Agreement and Privacy Policy.

Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website. See our Privacy Policy and User Agreement for details.

5.
nite Integration3. The Area Under a Curve4. Final Quiz Solutions to Exercises Solutions to Quizzes The full range of these packages and some instructions, should they be required, can be obtained from our web page Mathematics Support Materials.

14.
nite integral of the function f (x) with respect tox from a to b to be  b b f (x)dx = F (x) = F (b) F (a) ; a awhere F (x) is the anti-derivative of f (x). We call a and b the lowerand upper limits of integration respectively. The function being inte-grated, f (x), is called the integrand. Note the minus sign!Note integration constants are not written in de

23.
Section 3: The Area Under a Curve 7Consider the area, A, under the curve, y = f (x). If we increasethe value of x by x, then the increase in area, A, is approximately y A = y x A A = y : x Here we approximate the y area of the thin strip by a rectangle of width x and height y . In the limit A A as the strips become thin, x x+x x x 3 0, this means: 0 dA A = lim = y: dx x 30 xThe function (height of the curve) is the derivative of the area andthe area below the curve is an anti-derivative or integral ofthe function.N.B. so far we have assumed that y = f (x) lies above the x axis.

39.
Section 4: Final Quiz 154. Final QuizBegin Quiz Choose the solutions from the options given. 1. What is the area under the curve of the following positive function y = 10x4 + 3x2 between x = 1 and x = 2? (a) 75 ; (b) 53 ; (c) 69 ; (d) 57 : 2. What is the de

52.
nite integral, A y1 (x)dx, must be negative. Thisis because the function y1 (x) is negative for all values of x between Aand B . The area is all below the x axis.Click on the green square to return £